Lecture 02: One Period Model

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Lecture 02 One Period Model
Prof. Markus K. Brunnermeier
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Overview

• Securities Structure
• Arrow-Debreu securities structure
• Redundant securities
• Market completeness
• Completing markets with options
• Pricing (no arbitrage, state prices, SDF, EMM )
• Optimization and Representative Agent(Pareto
  efficiency, Welfare Theorems, )

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The Economy
s1

• State space (Evolution of states)
• Two dates t0,1
• S states of the world at time t1
• Preferences
• U(c0, c1, ,cS)
• (slope of
  indifference curve)
• Security structure
• Arrow-Debreu economy
• General security structure

s2
0

sS
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Security Structure

• Security j is represented by a payoff vector
• Security structure is represented by payoff
  matrix
• NB. Most other books use the inverse of X as
  payoff matrix.

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Arrow-Debreu Security Structure in R2
One A-D asset e1 (1,0)
This payoff cannot be replicated!
c2
Payoff Space ltXgt
c1
) Markets are incomplete
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Arrow-Debreu Security Structure in R2
Add second A-D asset e2 (0,1) to e1 (1,0)
c2
c1
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Arrow-Debreu Security Structure in R2
Add second A-D asset e2 (0,1) to e1 (1,0)
c2
Payoff space ltXgt
c1
Any payoff can be replicated with two A-D
securities
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Arrow-Debreu Security Structure in R2
Add second asset (1,2) to
c2
Payoff space ltXgt
c1
New asset is redundant it does not enlarge the
payoff space
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Arrow-Debreu Security Structure

• S Arrow-Debreu securities
• each state s can be insured individually
• All payoffs are linearly independent
• Rank of X S
• Markets are complete

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General Security Structure
Only bond
Payoff space ltXgt
c2
c1
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General Security Structure
Only bond xbond (1,1)
Payoff space ltXgt
cant be reached
c2
c1
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General Security Structure
Add security (2,1) to bond (1,1)
c2
c1
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General Security Structure
Add security (2,1) to bond (1,1)
c2

• Portfolio of
• buy 3 bonds
• sell short 1 risky asset

c1
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General Security Structure
c2
Payoff space ltXgt
c1
Two asset span the payoff space
Market are complete with security
structure Payoff space coincides with payoff
space of
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General Security Structure

• Portfolio vector h 2 RJ (quantity for each
  asset)
• Payoff of Portfolio h is åj hj xj hX
• Asset span
• ltXgt is a linear subspace of RS
• Complete markets ltXgt RS
• Complete markets if and only if rank(X) S
• Incomplete markets rank(X) lt S
• Security j is redundant if xj hX with hj0

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General Security Structure

• Price vector p 2 RJ of asset prices
• Cost of portfolio h,
• If pj ¹ 0 the (gross) return vector of asset j is
  the vector

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Options to Complete the Market
Stocks payoff
Introduce call options with final payoff at T
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Options to Complete the Market
Together with the primitive asset we obtain
Homework check whether this markets are complete.
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Overview

• Securities Structure (AD securities, Redundant
  securities, completeness, )
• Pricing
• LOOP, No arbitrage and existence of state prices
• Market completeness and uniqueness of state
  prices
• Pricing kernel q
• Three pricing formulas (state prices, SDF, EMM)
• Recovering state prices from options
• Optimization and Representative Agent(Pareto
  efficiency, Welfare Theorems, )

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Pricing

• State space (evolution of states)
• (Risk) preferences
• Aggregation over different agents
• Security structure prices of traded securities
• Problem
• Difficult to observe risk preferences
• What can we say about existence of state prices
  without assuming specific utility functions for
  all agents in the economy

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Vector Notation

• Notation y,x 2 Rn
• y x , yi xi for each i1,,n.
• y gt x , y x and y ¹ x.
• y gtgt x , yi gt xi for each i1,,n.
• Inner product
• y x åi yx
• Matrix multiplication

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Three Forms of No-ARBITRAGE

1. Law of one price (LOOP)If hX kX then p h
   p k.
2. No strong arbitrageThere exists no portfolio h
   which is a strong arbitrage, that is hX 0 and
   p h lt 0.
3. No arbitrageThere exists no strong arbitrage
   nor portfolio k with k X gt 0 and p k 0.

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Three Forms of No-ARBITRAGE

• Law of one price is equivalent to every
  portfolio with zero payoff has zero price.
• No arbitrage ) no strong arbitrage No strong
  arbitrage ) law of one price

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Pricing

• Define for each z 2 ltXgt,
• If LOOP holds q(z) is a single-valued and linear
  functional. Conversely, if q is a linear
  functional defined in ltXgt then the law of one
  price holds.

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Pricing

• LOOP ) q(hX) p h
• A linear functional Q in RS is a valuation
  function if Q(z) q(z) for each z 2 ltXgt.
• Q(z) q z for some q 2 RS, where qs Q(es),
  and es is the vector with ess 1 and esi 0 if
  i ¹ s
• es is an Arrow-Debreu security
• q is a vector of state prices

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State prices q

• q is a vector of state prices if p X q, that
  is pj xj q for each j 1,,J
• If Q(z) q z is a valuation functional then q
  is a vector of state prices
• Suppose q is a vector of state prices and LOOP
  holds. Then if z hX LOOP implies that
• Q(z) q z is a valuation functional , q is a
  vector of state prices and LOOP holds

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State prices q
p(1,1) q1 q2 p(2,1) 2q1 q2 Value of
portfolio (1,2) 3p(1,1) p(2,1) 3q1
3q2-2q1-q2 q1 2q2
c2
q2
c1
q1
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The Fundamental Theorem of Finance

• Proposition 1. Security prices exclude arbitrage
  if and only if there exists a valuation
  functional with q gtgt 0.
• Proposition 2. Let X be an J ? S matrix, and p
  2 RJ. There is no h in RJ satisfying h p 0,
  h X 0 and at least one strict inequality if,
  and only if, there exists a vector q 2 RS with
  q gtgt 0 and p X q.
• No arbitrage , positive state prices

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Multiple State Prices q Incomplete Markets
What state prices are consistent
with p(1,1)? p(1,1) q1 q2
One equation two unknowns q1, q2
There are (infinitely) many.
e.g. if p(1,1).9 q1 .45, q2
.45 or q1 .35, q2
.55
bond (1,1) only
c2
Payoff space ltXgt
p(1,1)
q2
c1
q1
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q
complete markets
ltXgt
q
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pXq
q
incomplete markets
ltXgt
q
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pXqo
q
incomplete markets
ltXgt
qo
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Multiple q in incomplete markets
c2
ltXgt
pXq
q
qo
q
v
c1
Many possible state price vectors s.t. pXq. One
is special q - it can be replicated as a
portfolio.
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Uniqueness and Completeness

• Proposition 3. If markets are complete, under no
  arbitrage there exists a unique valuation
  functional.
• If markets are not complete, then there exists v
  2 RS with 0 Xv. Suppose there is no arbitrage
  and let q gtgt 0 be a vector of state prices. Then
  q a v gtgt 0 provided a is small enough, and p
  X (q a v). Hence, there are an infinite number
  of strictly positive state prices.

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The Three Asset Pricing Formulas

• State prices pj ås qs xsj
• Stochastic discount factor pj Emxj
• Martingale measure pj 1/(1rf) Ep xj
• (reflect risk aversion by
• over(under)weighing the bad(good) states!)

xj1
m1
m2
xj2
m3
xj3

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Stochastic Discount Factor

• That is, stochastic discount factor ms qs/ps
  for all s.

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Stochastic Discount Factor
shrink axes by factor
ltXgt
m
m
c1
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Equivalent Martingale Measure

• Price of any asset
• Price of a bond

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The Three Asset Pricing Formulas

• State prices pj ås qs xsj
• Stochastic discount factor pj Emxj
• Martingale measure pj 1/(1rf) Ep xj
• (reflect risk aversion by
• over(under)weighing the bad(good) states!)

xj1
m1
m2
xj2
m3
xj3

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specify Preferences Technology
observe/specify existing Asset Prices

• evolution of states
• risk preferences
• aggregation

NAC/LOOP
NAC/LOOP
State Prices q (or stochastic discount
factor/Martingale measure)
absolute asset pricing
relativeasset pricing
LOOP
derivePrice for (new) asset
derive Asset Prices
Only works as long as market completeness
doesnt change
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Recovering State Prices from Option Prices

• Suppose that ST, the price of the underlying
  portfolio (we may think of it as a proxy for
  price of market portfolio), assumes a
  "continuum" of possible values.
• Suppose there are a continuum of call options
  with different strike/exercise prices ) markets
  are complete
• Let us construct the following portfolio for
  some small positive number egt0,
• Buy one call with
• Sell one call with
• Sell one call with
• Buy one call with .

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Recovering State Prices (ctd.)
Figure 8-2 Payoff Diagram Portfolio of Options
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Recovering State Prices (ctd.)

• Let us thus consider buying 1/e units of the
  portfolio. The total payment, when
  , is , for any choice
  of e. We want to let , so as to
  eliminate the payments in the ranges
  and
• .The value of 1/e units of this portfolio
  is 

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Taking the limit e ! 0
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Recovering State Prices (ctd.)
Evaluating following cash flow

• The value today of this cash flow is 

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(No Transcript)
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specify Preferences Technology
observe/specify existing Asset Prices

• evolution of states
• risk preferences
• aggregation

NAC/LOOP
NAC/LOOP
State Prices q (or stochastic discount
factor/Martingale measure)
absolute asset pricing
relativeasset pricing
LOOP
derivePrice for (new) asset
derive Asset Prices
Only works as long as market completeness
doesnt change
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Overview

• Securities Structure (AD securities, Redundant
  securities, completeness, )
• Pricing (no arbitrage, state prices, SDF, EMM )
• Optimization and Representative Agent
• Marginal Rate of Substitution (MRS)
• Pareto Efficiency
• Welfare Theorems
• Representative Agent Economy

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Representation of Preferences

• A preference ordering is (i) complete, (ii)
  transitive, (iii) continuous and (iv) relatively
  stable can be represented by a utility function,
  i.e. (c0,c1,,cS) Â (c0,c1,,cS)
• , U(c0,c1,,cS) gt U(c0,c1,,cS) (more on
  risk preferences in next lecture)

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Agents Optimization

• Consumption vector (c0, c1) 2 R x RS
• Agent i has Ui R x RS ! R
  endowments (e0,e1) 2 R x RS
• Ui is quasiconcave c Ui(c) v is convex for
  each real v
• Ui is concave for each 0 a 1, Ui (a c (1-
  a)c) a Ui (c) (1-a) Ui (c)
• Ui/ c0 gt 0, Ui/ c1 gtgt0

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Agents Optimization

• Portfolio consumption problem

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Agents Optimization
For time separable utility function
and expected utility function (later more)
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Agents Optimization

• To sum up
• Proposition 3 Suppose c gtgt 0 solves problem.
  Then there exists positive real numbers l, m1, ,
  mS, such that
• The converse is also true.
• The vector of marginal rate of substitutions
  MRSs,0 is a (positive) state price vector.

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Suppose cs is fixed since it cant be traded
cAs
cA0, cA1U(cA0, cA1)U(e)
slope
As indifference curve
e
Mr. A
cA0
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Ms. B
cB0
e
cBs
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Ms. B
cB0
cAs
eA, eB
cBs
Mr. A
cA0
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Ms. B
cB0
cAs
1
q
eA, eB
cBs
Mr. A
cA0
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Set of PO allocations (contract curve)
Ms. B
cB0
cAs
eA, eB
cBs
Mr. A
cA0
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Pareto Efficiency

• Allocation of resources such that
• there is no possible redistribution such that
• at least one person can be made better off
• without making somebody else worse off
• Note
• Allocative efficiency ? Informational efficiency
• Allocative efficiency ? fairness

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Set of PO allocations (contract curve) MRSA MRSB
Ms. B
cB0
cAs
eA, eB
cBs
Mr. A
cA0
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Welfare Theorems

• First Welfare Theorem. If markets are complete,
  then the equilibrium allocation is Pareto
  optimal.
• State price is unique q. All MRSi(c) coincide
  with unique state price q.
• Second Welfare Theorem. Any Pareto efficient
  allocation can be decentralized as a competitive
  equilibrium.

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Representative Agent Complete Markets

• Aggregation Theorem 1 Suppose
• markets are complete
• Then asset prices in economy with many agents
  are identical to an economy with a single
  agent/planner whose utility is
• U(c) åk ak
  uk(c), where ak is the welfare weights of agent
  k.and the single agent consumes the aggregate
  endowment.

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Representative Agent HARA utility world

• Aggregation Theorem 2 Suppose
• riskless annuity and endowments are tradeable.
• agents have common beliefs
• agents have a common rate of time preference
• agents have LRT (HARA) preferences with RA (c)
  1/(AiBc ) ) linear risk sharing rule
• Then asset prices in economy with many agents
  are indentical to a single agent economy with
  HARA preferences with RA(c) 1/(åi Ai B).

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Overview

1. Securities Structure (AD securities, Redundant
   securities, completeness, )
2. Pricing (no arbitrage, state prices, SDF, EMM )
3. Optimization and Representative Agent(Pareto
   efficiency, Welfare Theorems, )

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Extra Material

• Follows!

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Portfolio restrictions

• Suppose that there is a short-sale restriction
• where
• is a convex set
• d
• is a convex set
• F or z 2 let (cheapest
  portfolio replicating z)

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Restricted/Limited Arbitrage

• An arbitrage is limited if it involves a short
  position in a security
• In the presence of short-sale restrictions,
  security prices exclude (unlimited) arbitrage
  (payoff 1 ) if, and only if, here exists a q gtgt 0
  such that
• Intuition q MRSi from optimization problem
• some agents wished they could
  short-sell asset

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Portfolio restrictions (ctd.)

• As before, we may define Rf 1 / ås qs , and
  ps can be interpreted as risk-neutral
  probabilities
• Rf pj Ep xj, with if
• 1/Rf is the price of a risk-free security that is
  not subject to short-sale constraint.

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Portfolio restrictions (ctd.)

• Portfolio consumption problem
• max over c0, c1, h utility function u(c0,c1)
• subject to (i) 0 c0 w0 p h
• (ii) 0 c1 w1 h X
• Proposition 4 Suppose cgtgt0 solves problem s.t.
  hj bj for . Then there exists
  positive real numbers l, m1, m2, ..., mS, such
  that
• Ui/ c0 (c) l
• Ui/ c0 (c) (m1, ...,mS)
• s
• s
• The converse is also true.

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FOR LATER USE Stochastic Discount Factor

• That is, stochastic discount factor ms qs/ps
  for all s.